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Verify if (6, 4) is a Solution to the Linear System $\\begin{cases} x + y = 10 \\\\ 2x + 3y = 24 \\end{cases}$
Mathematics
Grade 8
Question Content
Test the given solution (6, 4) to confirm it works for the system: $\\begin{cases} x + y = 10 \\\\ 2x + 3y = 24 \\end{cases}$
Correct Answer
The solution (6, 4) is valid for the system of equations.
Detailed Solution Steps
1
Step 1: Identify the values of x and y from the given solution. Here, $x=6$ and $y=4$.
2
Step 2: Substitute $x=6$ and $y=4$ into the first equation $x + y = 10$. Calculate the left-hand side: $6 + 4 = 10$, which equals the right-hand side, so the solution satisfies the first equation.
3
Step 3: Substitute $x=6$ and $y=4$ into the second equation $2x + 3y = 24$. Calculate the left-hand side: $2(6) + 3(4) = 12 + 12 = 24$, which equals the right-hand side, so the solution satisfies the second equation.
4
Step 4: Since the solution satisfies both equations in the system, it is a valid solution.
Knowledge Points Involved
1
Solution Verification for Systems of Linear Equations
This refers to substituting the given ordered pair (x, y) into each equation in the system. If the left-hand side equals the right-hand side for every equation, the ordered pair is a valid solution for the system. It is used to confirm if a proposed solution works for all equations in the system simultaneously.
2
Substitution in Algebraic Equations
This is the process of replacing variables in an equation with their given numerical values. It is a fundamental algebraic technique used to evaluate expressions, verify solutions, and solve equations. In this context, it is used to check if the values of x and y make each equation true.
3
Systems of Linear Equations in Two Variables
A set of two linear equations with two variables (usually x and y) that share a common solution set. A valid solution must satisfy both equations at the same time, representing the intersection point of the two lines when graphed.
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